The topological snake lemma and corona algebras
If \(B\) is a \(C^*\)-algebra then the multiplier algebra of \(B\) is denoted by \({\mathcal M}B\) and the Corona algebra is denoted by \({\mathcal Q}B={\mathcal M}B/B\). A *-isomorphism \(f: B\to B'\) is quasi-unital when there is a projection \(p\in{\mathcal M}B\) such that the closed linear span of \(f(B)B'\) has the form \(pB'\). The main result of the paper is the following. Suppose that \(B\) and \(B'\) are separable \(C^*\)-algebras and that \(f:B\to B'\) is a quasi-unital monomorphism, then the induced map \(\overline f:{\mathcal Q}B\to{\mathcal Q}B'\) is a monomorphism, too. This result is based on the new \(C^*\)-algebra version of the so-named snake lemma from homological algebra.
scientific article; zbMATH DE number 1348672